Rigorous Proof for Riemann Hypothesis That Rigidly Comply with Principle of Maximum Density for Integer Number Solutions

viXra manuscript No. (will be inserted by the editor)

Rigorous proof for Riemann hypothesis that rigidly comply with Principle of Maximum Density for Integer Number Solutions

John Ting

June 17, 2021

Abstract The 1859 Riemann hypothesis conjectured all nontrivial zeros in Riemann zeta function are uniquely located on sigma = 1/2 critical line. Derived from Dirichlet eta function [proxy for Riemann zeta function] are, in chronological order, simplified Dirichlet eta function and Dirichlet Sigma-Power Law. Computed Zeroes from the former uniquely occur at sigma = 1/2 resulting in total summation of fractional exponent (–sigma) that is twice present in this function to be integer –1. Computed Pseudo-zeroes from the later uniquely occur at sigma = 1/2 resulting in total summation of fractional exponent (1 – sigma) that is twice present in this law to be integer 1. All nontrivial zeros are, respectively, obtained directly and indirectly as one specific type of Zeroes and Pseudo-zeroes only when sigma = 1/2. Thus, it is proved that Riemann hypothesis is true whereby this function and law must rigidly comply with Principle of Maximum Density for Integer Number Solutions. Keywords Dirichlet Sigma-Power Law · Pseudo-zeroes · Riemann hypothesis · Zeroes Mathematics Subject Classification (2010) 11M26 · 11A41

Contents

1 Introduction ...... 1 2 Notation and sketch of the proof ...... 2 3 The Completely Predictable and Incompletely Predictable entities ...... 10 4 The exact and inexact Dimensional analysis homogeneity for Equations ...... 11 5 Gauss Areas of Varying Loops and Principle of Maximum Density for Integer Number Solutions ...... 11 6 Simple observation on Shift of Varying Loops in ζ(σ + ıt) Polar Graph and Principle of Equidistant for Multiplicative Inverse...... 14 7 Riemann zeta function, Dirichlet eta function, simpliﬁed Dirichlet eta function and Dirichlet Sigma-Power Law . . . . . 18 8 Conclusions ...... 20 Declaration of interests ...... 20 Acknowledgment ...... 20 References ...... 20

1 Introduction

Riemann hypothesis is an intractable open problem in Number theory that was proposed in 1859 by famous German mathematician Bernhard Riemann (September 17, 1826 - July 20, 1866). This hypothesis conjectured 1 all nontrivial zeros in Riemann zeta function are uniquely located on σ = 2 critical line. By applying Euler

Corresponding Author E-mail: [email protected] (Alma Mater: University of Queensland) Dr. John Yuk Ching Ting, Dental and Medical Surgery, 729 Albany Creek Road, Albany Creek QLD 4035, Australia 2 Ting

1 2 3 Fig. 1: INPUT for σ = 2 , 5 , and 5 . ζ(s) has CIS of CP trivial zeros located at σ = all negative even numbers 1 and [conjectured] CIS of IP NTZ located at σ = 2 given by various t values. formula to Dirichlet eta function [proxy for Riemann zeta function], we obtain simplified Dirichlet eta function 1 whereby its computed Zeroes uniquely occur at σ = 2 resulting in total summation of fractional exponent (–σ) that is twice present in this function to be integer –1. Dirichlet Sigma-Power Law is the solution from performing integration on simplified Dirichlet eta function whereby its computed Pseudo-zeroes uniquely occur 1 at σ = 2 resulting in total summation of fractional exponent (1 – σ) that is twice present in this law to be integer 1. All nontrivial zeros are, respectively, obtained directly and indirectly as one specific type of Zeroes 1 and Pseudo-zeroes only when σ = 2 . Then [non-existent] virtual nontrivial zeros and virtual Pseudo-nontrivial zeros cannot be obtained directly and indirectly as a type of virtual Zeroes and virtual Pseudo-zeroes when 1 σ 6= 2 . Based on Theorem 1 – 3 with associated Lemma 1 – 2 below, it is proved that Riemann hypothesis is true whereby this function and law must rigidly comply with Principle of Maximum Density for Integer Number Solutions. In addition, they also serendipitously obey trigonometric identities and manifest Principle of Equidistant for Multiplicative Inverse.

2 Notation and sketch of the proof

Notation. CFS: countable finite set CIS: countable infinite set UIS: uncountable infinite set CP: Completely Predictable – see section 3 on definition for CP entities IP: Incompletely Predictable – see section 3 on definition for IP entities DA: Dimensional analysis – see section 4 on definitions for exact and inexact DA homogeneity NTZ: nontrivial zeros ζ(s): Riemann zeta function η(s): Dirichlet eta function sim-η(s): simplified-Dirichlet eta function DSPL: Dirichlet Sigma-Power Law Symbolically named after German mathematician Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859), the word “Law” utilized in DSPL represent a convenient terminology to describe this function – viz, there is resemblance to Zipf’s law via power law functions in σ from s = σ + it being exponent of a power function as similar format to nσ , logarithm scale use, and harmonic ζ(s) series connection. Respectively, we use the terms 1 Zeroes (as three types of Gram points) and Pseudo-zeroes (as three types of Pseudo-Gram points) at σ = 2 to collectively refer to corresponding f (n)’s and F(n)’s x-axis intercept points, y-axis intercept points and Origin intercept points. Respectively, we use the terms virtual Zeroes (as two types of virtual Gram points) and virtual 1 Pseudo-zeroes (as two types of virtual Pseudo-Gram points) at σ 6= 2 to collectively refer to corresponding f (n)’s and F(n)’s x-axis intercept points and y-axis intercept points [with absent Origin intercept points]. Riemann hypothesis and closely related two types of Gram points 3

Geometrical and mathematical definitions for Gram points and virtual Gram points. Figure 1 depicts complex variable s (= σ ±ıt) as INPUT with x-axis denoting real part Re{s} associated with σ, and y-axis denoting 1 1 1 imaginary part Im{s} associated with t. Then critical line: σ = 2 ; non-critical lines: σ 6= 2 viz, 0 < σ < 2 and 1 1 1 2 < σ < 1; and critical strip: 0 < σ < 1. Both unique σ = 2 value and non-unique σ 6= 2 values ∈ Set all σ values whereby Set all σ values = σ | σ is a real number, and 0 < σ < 1. With including its complex conjugate, s = σ ± it is present in all our well-defined f(n) and F(n) whereby these are continuous [complex] functions always defined for any arbitrarily chosen intervals [a,b]. Given f(n) = 0 and F(n) = 0 as relevant derived equations, they dependently generate corresponding types of IP entities. All these IP entities will inherently be correctly assigned to appropriate independent mutually exclusive CIS constituted by t values as transcendental numbers except for the first Gram[y=0] point (and first virtual Gram[y=0] point) given by t = 0. Origin intercept points, x-axis intercept points and y-axis intercept points are geometrical definitions for IP entities of Gram[x=0,y=0] points, Gram[y=0] points and Gram[x=0] points. These geometrical definitions are equivalent to corresponding 1 mathematical definitions given by the relevant equations below for σ = 2 : The Origin intercept points consisting of Gram[x=0,y=0] points or NTZ are computed directly from equa- 1 tions η(s) = 0 and sim-η(s) = 0; and indirectly from equation DSPL = 0 when σ = 2 . The x-axis intercept points consisting of Gram[y=0] points or (traditional) ‘usual’ Gram points are computed directly from equa- 1 tion Gram[y=0] points-sim-η(s) = 0; and indirectly from equation Gram[y=0] points-DSPL = 0 when σ = 2 . The y-axis intercept points consisting of Gram[x=0] points are computed directly from equation Gram[x=0] 1 points-sim-η(s) = 0; and indirectly from equation Gram[x=0] points-DSPL = 0 when σ = 2 . Relevant functions and equations are unique mathematical objects that can be classified as three types of inifinite series: Harmonic series, Alternating harmonic series or Alternating series with trigonometric terms. We perform crucial de novo analysis on these functions and equations by noting their manifested intrinsic properties. Without loss of validity in our correct and complete set of mathematical arguments, we adopt the convention of providing focused analysis predominantly on appropriately chosen Alternating series with trigonometric terms throughout our presentation. The complex f(n) ζ(s) is a Harmonic series that does not converge in the critical strip. The complex f(n) η(s) is an Alternating harmonic series that converge in the critical strip. Then η(s) must act as proxy function for ζ(s) in this strip. Derived as Euler formula application to η(s) is the complex f(n) sim-η(s), and derived as R sim − η(s)dn is the complex F(n) DSPL. Both sim-η(s) and DSPL are Alternating series with trigonometric terms that converge in the critical strip. The f(n) η(s) will converge infinitely often to a zero value as η(s) = 0 equation giving rise to all NTZ or Gram[x=0,y=0] points. This event will only happen when η(s) is substituted with one unique σ value which is 1 conjectured to be σ = 2 by Riemann hypothesis. Being an Alternating harmonic series, we inherently cannot derive valid functions to obtain corresponding equations Gram[y=0] points-η(s) = 0 and Gram[x=0] points-η(s) = 0 that will enable mathematical computations of Gram[y=0] points as x-axis intercept points and Gram[x=0] points as y-axis intercept points. Then, computed Zeroes are mathematically defined as η(s) = 0 and sim-η(s) 1 = 0 when variable σ = 2 ; computed virtual Zeroes are mathematically defined as η(s) 6= 0 and sim-η(s) = 0 1 1 when variable σ 6= 2 ; computed Pseudo-Zeroes are mathematically defined as DSPL = 0 when variable σ = 2 ; 1 and computed virtual Pseudo-zeroes are mathematically defined as DSPL = 0 when variable σ 6= 2 . For 0 ≤ δ ≤ 1, let f (n) = sin(n)±δ and f (n) = cos(n)±δ represent two [simple] trigonometric functions which are periodic transcendental-type functions. Both sin(n) ± δ = 0 and cos(n) ± δ = 0 as equations will generate infinitely many CP x-axis intercept points (Zeroes) for any given values of δ. This will additionally include the solitary Origin intercept point (Zero) obtained from sin(n)±δ = 0 when δ = 0. For both sin(n)±δ and cos(n) ± δ, only when δ = 0 will their progressive / cummulative Areas Above the horizontal axis be overall identical to Areas Below the horizontal axis. Otherwise, these mentioned Areas will not be overall identical to each other when δ 6= 0. We now provide analogical reasoning for existence of infinitely many 1 substituted σ values (including σ = 2 ) that will all contribute to two conditions sim-η(s) = 0 and DSPL = 0 1 being satisfied while simultaneously giving rise to (i) IP Zeroes and IP Pseudo-zeroes [when σ = 2 ], and (ii) 1 IP virtual Zeroes and IP virtual Pseudo-zeroes [when σ 6= 2 ]. With (complex) sine and/or cosine terms present 1 in f(n) sim-η(s) and F(n) DSPL also being periodic transcendental-type functions, we intuitively deduce σ = 2 1 and σ 6= 2 must act as the analogical equivalence of δ = 0 and δ 6= 0. This deduction allows intuitive and valid 4 Ting

1 Fig. 2: OUTPUT for σ = 2 as Gram points. This Figure represents schematically depicted polar graph of 1 1 ζ( 2 + ıt) plotted along critical line for real values of t running from 0 to 34, horizontal axis: Re{ζ( 2 + ıt)}, 1 and vertical axis: Im{ζ( 2 + ıt)}. There are presence of Origin intercept points.

2 Fig. 3: OUTPUT for σ = 5 as virtual Gram points. Varying Loops are shifted to left of Origin with horizontal 2 2 axis: Re{ζ( 5 + ıt)}, and vertical axis: Im{ζ( 5 + ıt)}. There is total absence of Origin intercept points.

3 3 Fig. 4: OUTPUT for σ = 5 as virtual Gram points with horizontal axis: Re{ζ( 5 + ıt)}, and vertical axis: 3 Im{ζ( 5 + ıt)}. Varying Loops are shifted to right of Origin. There is total absence of Origin intercept points. explanations for our two conditions to be satisfied by the infinitely many substituted σ values. Consequently, we must rigorously prove additional property of sim-η(s) and DSPL that they will characteristically, inevitably 1 and uniquely comply with Principle of Maximum Density for Integer Number Solutions only when σ = 2 with this Principle signifying complete presence of NTZ in sim-η(s) or Pseudo-NTZ in DSPL as one unique type 1 of Gram points or Pseudo-Gram points [which will otherwise be totally absent when σ 6= 2 ]. Figures 2, 3 and 4 are ζ(σ + ıt) Polar Graphs with x-axis denoting real part Re{ζ(s)} and y-axis denoting imaginary part Im{ζ(s)} generated by ζ(s)’s output as real values of t running from 0 to 34. There are infinite types-of-spirals (Varying Loops) possibilities associated with each σ value arising from all infinite σ values 1 in 0 < σ < 1 critical strip whereby the unique and solitary σ = 2 value that denote critical line is located in Riemann hypothesis and closely related two types of Gram points 5 this strip. We observe that Figures 3 and 4 show associated shifts of Varying Loops that manifest Principle of Equidistant for Multiplicative Inverse - see Lemma 2 from section 6. From observing Figure 2, we can 1 geometrically define NTZ (or Gram[x=0,y=0] points) as Origin intercept points occurring when σ = 2 . Then, two remaining types of Gram points as part of continuous Varying Loops are consequently defined as x-axis 1 intercept points and y-axis intercept points occurring when σ = 2 . From Remark 6, corresponding three types of F(n)’s Pseudo-zeroes or Pseudo-Gram points and two types of F(n)’s virtual Pseudo-zeroes or virtual Pseudo-Gram points can be precisely converted to three types of f(n)’s Zeroes or Gram points and two types of f(n)’s virtual Zeroes or virtual Gram points. Then, Statement (I) – (IV) 1 below are valid whereby σ = 2 ’s derived entities from (III) can be precisely converted to those from (I), and 1 σ 6= 2 ’s derived virtual entities from (IV) can be precisely converted to those from (II): 1 (I) The f(n)’s Zeroes at σ = 2 [directly] equates to three types of Gram points. 1 (II) The f(n)’s virtual Zeroes at σ 6= 2 [directly] equates to two types of virtual Gram points. 1 (III) The F(n)’s Pseudo-zeroes at σ = 2 [indirectly] equates to three types of Gram points. 1 (IV) The F(n)’s virtual Pseudo-zeroes at σ 6= 2 [indirectly] equates to two types of virtual Gram points. Remark 1. Of particular relevance to Riemann hypothesis, we mathematically deduce from above materials that the f(n)’s NTZ or Gram[x=0,y=0] points as one type of Gram points will conjecturally only exist at unique 1 1 σ = 2 critical line [but not at non-unique σ 6= 2 non-critical lines]. This can be equivalently stated as: The F(n)’s Pseudo-NTZ or Pseudo-Gram[x=0,y=0] points as one type of Pseudo-Gram points will conjecturally 1 1 only exist at unique σ = 2 critical line [but not at non-unique σ 6= 2 non-critical lines]. A line consists of infinitely many points. Graphically, the Origin is a zero-dimensional point; x-axis or horizontal axis and y-axis or vertical axis are one-dimensional lines. Remark 2. From Figures 3 and 4, we note Origin intercept points as Gram[x=0,y=0] points or NTZ cannot 1 exist when σ 6= 2 . From Figure 2, we note Origin intercept points as Gram[x=0,y=0] points or NTZ can 1 only exist when σ = 2 . Of particular relevance to Riemann hypothesis, we geometrically deduce sim-η(s) as periodic transcendental-type function must only contain one solitary σ-valued type of Origin intercept points 1 (when σ = 2 for Gram[x=0,y=0] points or NTZ as conjectured by Riemann hypothesis) but infinitely many 1 different σ-valued types of x-axis intercept points and y-axis intercept points (when σ = 2 for Gram[y=0] 1 points and Gram[x=0] points as well as σ 6= 2 for virtual Gram[y=0] points and virtual Gram[x=0] points). We can conjure up an equivalent statement for DSPL as periodic transcendental-type function whereby we simply replace NTZ and (virtual) Gram points with their counterparts Pseudo-NTZ and (virtual) Pseudo-Gram points. 1 We can now propose Theorem 1 (with σ = 2 connoting exact DA homogeneity) and its corollary Theorem 1 2 (with σ 6= 2 connoting inexact DA homogeneity) to fully represent Remark 1 and Remark 2. Their successful proofs will firstly, denote rigorous proof for Riemann hypothesis that involves conjecture on location of NTZ as one type of Gram points [viz, Origin intercept points] and secondly, provide precise explanations for remaining two types of Gram points [viz, x-axis intercept points and y-axis intercept points]. In addition, we incorporate Theorem 3 on rigid compliance by sim-η(s) and DSPL with Principle of Maximum Density for 1 Integer Number Solutions whereby its successful proof will only eventuate when σ = 2 .

Theorem 1. Rigidly complying with exact DA homogeneity, f(n) sim-η(s) and F(n) DSPL as relevant 1 equations can incorporate three types of Gram points and Pseudo-Gram points onto solitary σ = 2 critical line. 1 Proof. We note CIS of Gram[x=0,y=0] points or NTZ constitutes one type of Gram points when σ = 2 thus fully supporting Riemann hypothesis to be true. The preceding sentence is also valid when we replace Gram[x=0,y=0] points, NTZ and Gram points with corresponding Pseudo-Gram[x=0,y=0] points, Pseudo-NTZ and Pseudo-Gram points. Respectively, conveniently defined term of exact DA homogeneity denote [exact] integer −1 and 1 derived from ∑(all fractional exponents) = 2(−σ) and 2(1−σ). These act as surrogate markers 1 1 in sim-η(s) and DSPL on [solitary] σ = 2 situation. Generated by relevant functions and laws when σ = 2 , the three types of Gram points are mathematically defined as equations sim-η(s) = 0, Gram[y=0] points-sim-η(s) = 0 and Gram[x=0] points-sim-η(s) = 0; and the three types of Pseudo-Gram points are mathematically defined as equations DSPL = 0, Gram[y=0] points-DSPL = 0 and Gram[x=0] points-DSPL = 0. They all correspond to relevant geometrically defined Origin intercept points, x-axis intercept points and y-axis intercept points. Thus, 6 Ting three types of IP Gram points [IP Zeroes] and IP Pseudo-Gram points [IP Pseudo-Zeroes] are mathematically 1 and geometrically defined to be located on σ = 2 critical line. Based solely on these definitive definitions, we can uniquely incorporate three types of IP Gram points [IP Zeroes] and IP Pseudo-Gram points [IP Pseudo- 1 zeroes] onto σ = 2 critical line. The proof is now complete for Theorem 12.

Theorem 2. Rigidly complying with inexact DA homogeneity, f(n) sim-η(s) and F(n) DSPL as relevant equations can incorporate two types of virtual Gram points and virtual Pseudo-Gram points onto infinitely 1 many σ 6= 2 non-critical lines. Proof. We note [non-existent] virtual Gram[x=0,y=0] points or virtual NTZ will not constitute one type 1 of [non-existent] virtual Gram points when σ 6= 2 thus also fully supporting Riemann hypothesis to be true. The preceding sentence is also valid when we replace virtual Gram[x=0,y=0] points, virtual NTZ and virtual Gram points with corresponding virtual Pseudo-Gram[x=0,y=0] points, virtual Pseudo-NTZ and virtual Pseudo-Gram points. Respectively, conveniently defined term of inexact DA homogeneity denote [inexact] fractional (non-integer) number 6= −1 and 6= 1 derived from ∑(all fractional exponents) = 2(−σ) and 2(1−σ). 1 These act as surrogate markers in sim-η(s) and DSPL on [infinitely many] σ 6= 2 situations. Generated by 1 relevant functions and laws when σ 6= 2 , the two types of virtual Gram points are mathematically defined as equations virtual Gram[y=0] points-sim-η(s) = 0 and virtual Gram[x=0] points-sim-η(s) = 0; and the two types of virtual Pseudo-Gram points are mathematically defined as equations virtual Gram[y=0] points-DSPL = 0 and virtual Gram[x=0] points-DSPL = 0. They all correspond to relevant geometrically defined x-axis intercept points and y-axis intercept points. Thus, two types of IP virtual Gram points [IP virtual Zeroes] and IP virtual Pseudo-Gram points [IP virtual Pseudo-Zeroes] are mathematically and geometrically defined to be 1 located on σ 6= 2 non-critical lines. Based solely on these definitive definitions, we can uniquely incorporate two types of IP virtual Gram points [IP virtual Zeroes] and IP virtual Pseudo-Gram points [IP virtual Pseudo- 1 zeroes] onto σ 6= 2 non-critical lines. The proof is now complete for Theorem 22.

1 1 Theorem 3. Conforming to the solitary σ = 2 critical line [and not the infinitely many σ 6= 2 non-critical 1 2 lines e.g. σ = 3 or 3 ] whereby σ forms part of relevant fractional exponents from base quantities (2n) and (2n-1) in sim-η(s) [as Riemann sum ∆n −→ 1 with variable n involving all integers ≥ 1] or DSPL [as definite integral ∆n −→ 0 with variable n involving all real numbers ≥ 1]; square roots of perfect squares [and not e.g. cube roots of perfect cubes or squared cube roots of perfect cubes] when applied to combined base quantities (2n) and (2n-1) in sim-η(s) or DSPL will generate the maximum number of integer solutions (constituted by all integers ≥ 1) that uniquely comply with Principle of Maximum Density for Integer Number Solutions while also manifesting Principle of Equidistant for Multiplicative Inverse. Proof. R sim-η(s)dn = DSPL. Whereas the two subsets of rational roots as integers and irrational roots as irrational numbers can be generated by combined base quantities (2n) and (2n-1) from sim-η(s) [with variable n involving all integers ≥ 1], so must these two exact same subsets be generated by combined base quantities (2n) and (2n-1) from DSPL [with variable n involving all real numbers ≥ 1]. Thus in sim-η(s) or DSPL, its computed CIS rational roots (subset) as integers [rational numbers] + computed CIS irrational roots (subset) as irrational numbers = computed CIS total roots. These two mutually exclusive subsets belong to UIS real 1 numbers. Using subset rational roots as integers at σ = 2 critical line, and by comparing and contrasting this 1 2 subset with [different] subset rational roots as integers at σ = 3 or 3 non-critical lines corollary situation; we will show that square roots of perfect squares [and not e.g. cube roots of perfect cubes or squared cube roots of perfect cubes] when applied to combined base quantities (2n) and (2n-1) from sim-η(s) or DSPL giving rise to maximum number of integer solutions (constituted by all integers ≥ 1) must uniquely comply with Princi- ple of Maximum Density for Integer Number Solutions (see Lemma 1 from section 5) while also manifesting Principle of Equidistant for Multiplicative Inverse (see Lemma 2 from section 6). We also apply concepts from elegant Gauss Circle Problem and Primitive Circle Problem into our supporting materials on this aptly-named Gauss Areas of Varying Loops to justifiably obtain correct and complete set of mathematical arguments that fully support Theorem 3. The proof is now complete for Theorem 32. Riemann hypothesis and closely related two types of Gram points 7

By conveniently employing only sim-η(s) for analysis here [with analysis using DSPL being equally valid], Theorem 1 and Theorem 2 above can also be insightfully combined as follows. Let Set G = all Gram points = Gram[x=0,y=0] points + Gram[y=0] points + Gram[x=0] points and Set vG = all virtual Gram points = virtual Gram[y=0] points + virtual Gram[x=0] points with virtual Gram[x=0,y=0] points = null set /0.We can apply inclusion-exclusion principle |G ∪ vG| = |G| + |vG| − |G ∩ vG| = |G| + |vG| because |G ∩ vG| = 0. Since exclusive presence of Gram points and absence of virtual Gram points on critical line denotes exclusive absence of Gram points and exclusive presence of virtual Gram points on non-critical lines; then Gram points and virtual Gram points as mutually exclusive entities must mathematically and geometrically be incorporated, respectively, onto unique (solitary) critical line and non-unique (inﬁnitely many) non-critical lines of sim-η(s). 1 2 Derived f(n) = 0 and F(n) = 0 equations – see σ = 2 and 5 representative examples given in [1], p. 1 27-28, 29-30 and section 4 below – comply with exact DA homogeneity at σ = 2 critical line and inexact 1 DA homogeneity at σ 6= 2 non-critical lines. NTZ are synonymous with Gram[x=0,y=0] points which is one type of Gram points. Together with Gram[y=0] points and Gram[x=0] points as remaining two types of Gram points, these three types of Gram points are intrinsically located in their complex equations (akin to Complex Containers) as IP entities whereby their actual location [but not actual positions] are intrinsically incorporated in these complex equations – see section 3 below. Eqs. (2.1), (2.3), (2.5), (2.6), (2.7) and (2.8) that comply with 1 1 exact DA homogeneity at σ = 2 all have fractional exponents 2 . Eqs. (2.2) and (2.4) that comply with inexact 2 2 3 DA homogeneity at σ = 5 have fractional exponents 5 in the former and 5 in the later that are mixed with 1 fractional exponents 2 .

∞ ∞ − 1 1 1 − 1 1 1 ∑ (2n) 2 2 2 cos(t ln(2n) + π) − ∑ (2n − 1) 2 2 2 cos(t ln(2n − 1) + π) = 0 (2.1) n=1 4 n=1 4 1 With exact DA homogeneity, Eq. (2.1) is f(n) sim-η(s) at σ = 2 that will incorporate all NTZ [as Zeroes]. There is total absence of (non-existent) virtual NTZ [as virtual Zeroes].

∞ ∞ − 2 1 1 − 2 1 1 ∑ (2n) 5 2 2 cos(t ln(2n) + π) − ∑ (2n − 1) 5 2 2 cos(t ln(2n − 1) + π) = 0 (2.2) n=1 4 n=1 4 2 With inexact DA homogeneity, Eq. (2.2) is f(n) sim-η(s) at σ = 5 that will incorporate all (non-existent) virtual NTZ [as virtual Zeroes]. There is total absence of NTZ [as Zeroes].

∞ 1 1 1 1 1 · (2n) 2 cos(t ln(2n) − π) − (2n − 1) 2 cos(t ln(2n − 1) − π) +C = 0 (2.3) 1 1 4 4 2 2 1 2 1 2 t + 4 1 With exact DA homogeneity, Eq. (2.3) is F(n) DSPL at σ = 2 that will incorporate all NTZ [as Pseudo- zeroes to Zeroes conversion]. There is total absence of (non-existent) virtual NTZ [as virtual Pseudo-zeroes to virtual Zeroes conversion].

∞ 1 3 1 3 1 · (2n) 5 cos(t ln(2n) − π) − (2n − 1) 5 cos(t ln(2n − 1) − π) +C = 0 (2.4) 1 1 4 4 2 2 9 2 1 2 t + 25 2 With inexact DA homogeneity, Eq. (2.4) is F(n) DSPL at σ = 5 that will incorporate all (non-existent) virtual NTZ [as virtual Pseudo-zeroes to virtual Zeroes conversion]. There is total absence of NTZ [as Pseudo- zeroes to Zeroes conversion].

∞ ∞ − 1 − 1 ∑ (2n) 2 sin(t ln(2n)) − ∑ (2n − 1) 2 sin(t ln(2n − 1)) = 0 (2.5) n=1 n=1 Eq. (2.5) can also be equivalently written as ∞ ∞ − 1 1 − 1 1 ∑ (2n) 2 cos(t ln(2n) − π) − ∑ (2n − 1) 2 cos(t ln(2n − 1) − π) = 0. n=1 2 n=1 2 8 Ting

1 With exact DA homogeneity, Eq. (2.5) is f(n) Gram[y=0] points-sim-η(s) at σ = 2 that will incorporate all Gram[y=0] points [as Zeroes]. There is total absence of virtual Gram[y=0] points [as virtual Zeroes].

∞ 1 1 1 1 2 − 1 · (2n) (cos(t ln(2n) − π) − cos(t ln(2n − 1) − π)) +C = 0 (2.6) 2 1 2 4 4 2(t + 4 ) 1 Eq. (2.6) can also be equivalently written as ∞ 1 1 3 3 2 1 · (2n) (cos(t ln(2n) + π) − cos(t ln(2n − 1) + π)) +C = 0. 2 1 2 4 4 2(t + 4 ) 1 1 With exact DA homogeneity, Eq. (2.6) is F(n) Gram[y=0] points-DSPL at σ = 2 that will incorporate all Gram[y=0] points [as Pseudo-zeroes to Zeroes conversion]. There is total absence of virtual Gram[y=0] points [as virtual Pseudo-zeroes to virtual Zeroes conversion].

∞ ∞ − 1 − 1 ∑ (2n) 2 cos(t ln(2n)) − ∑ (2n − 1) 2 cos(t ln(2n − 1)) = 0 (2.7) n=1 n=1 1 With exact DA homogeneity, Eq. (2.7) is f(n) Gram[x=0] points-sim-η(s) at σ = 2 that will incorporate all Gram[x=0] points [as Zeroes]. There is total absence of virtual Gram[x=0] points [as virtual Zeroes].

∞ 1 1 3 3 2 1 · (2n) (cos(t ln(2n) − π) − cos(t ln(2n − 1) − π)) +C = 0 (2.8) 2 1 2 4 4 2(t + 4 ) 1 1 With exact DA homogeneity, Eq. (2.8) is F(n) Gram[x=0] points-DSPL at σ = 2 that will incorporate all Gram[x=0] points [as Pseudo-zeroes to Zeroes conversion]. There is total absence of virtual Gram[x=0] points [as virtual Pseudo-zeroes to virtual Zeroes conversion]. We outline sim-η(s) as Eq. (2.2) and DSPL as Eq. (2.4) that comply with inexact DA homogeneity at 2 2 1 σ = 5 non-critical line (depicted by Figure 3) whereby σ = 5 [instead of σ = 2 ] is substituted into these two equations. Using [selective] trigonometric identity for linear combination of sine and cosine function whenever 1 applicable to relevant f(n) = 0 and F(n) = 0 equations, we outline exact DA homogeneity at σ = 2 critical line (depicted by Figure 2) for Gram[x=0,y=0] points (NTZ) as Eq. (2.1), Gram[y=0] points as Eq. (2.5) and Gram[x=0] points as Eq. (2.7). However, f(n) = 0 equations for Gram[y=0] points as Eq. (2.5) and Gram[x=0] 1 points as Eq. (2.7) with exact DA homogeneity at σ = 2 critical line are not amendable to treatments using trigonometric identity with implication that their corollary situation endowed with inexact DA homogeneity 1 1 at σ 6= 2 non-critical lines (depicted by Figures 3 and 4) will only manifest solitary [unmixed] 6= 2 fractional exponents. We will not provide [self-explanatory] examples of f(n) = 0 equations for this corollary situation. We arbitrarily chose single cosine wave with format Rcos(n ± α) where R is scaled amplitude and α is phase shift. For equations regarding NTZ, Gram[y=0] points and Gram[x=0] points; all their approximate Areas of Varying Loops ∝ precise Areas of Varying Loops with R validly treated as a proportionality factor. 1 1 1 2 − 2 We analyze f(n) = 0 and F(n) = 0 equations at σ = 2 critical line for NTZ situation where R = 2 (2n) or 1 − 1 1 1 1 1 2 2 (2n − 1) 2 in f(n)’s Eq. (2.1) and R = (2n) 2 or (2n − 1) 2 in F(n)’s Eq. (2.3). 1 1 1 1 2 2 1 2 2 2 1 2 2 t + 4 2 t + 4 Remark 3. Whereas for NTZ F(n) Eq. (2.3) that exactly represent precise Areas of Varying Loops and f(n) Eq. (2.1) [when interpreted as Riemann sum] that exactly represent approximate Areas of Varying Loops in a proportionate manner; so must the associated scaled amplitude R from Eq. (2.3) which is dependent on parameter t and Eq. (2.1) which is independent of parameter t represent [in a surrogate manner] corresponding precise and approximate Areas of Varying Loops in a proportionate manner. 1 We analyze f(n) = 0 equations [relevant to approximate Areas of Varying Loops] at σ = 2 critical line for Gram[y=0] points as Eq. (2.5) and Gram[x=0] points as Eq. (2.7) whereby we can validly designate R = − 1 − 1 (2n) 2 or (2n − 1) 2 as their assigned scaled amplitude and [unwritten] α = 0 as their assigned phase shift. 1 Relevant to precise Areas of Varying Loops at σ = 2 critical line for Gram[y=0] points F(n) Eq. (2.6) with R 1 1 1 1 1 1 2 2 2 = − 1 (2n) or − 1 (2n − 1) and Gram[x=0] points F(n) Eq. (2.8) with R = 1 (2n) 2 1 2 2 1 2 2 1 2 2 t + 4 2 t + 4 2 t + 4 Riemann hypothesis and closely related two types of Gram points 9

1 1 2 or 1 (2n − 1) , we observe the former R to be the negative of the later R. However, this observation 2 1 2 2 t + 4 is context-sensitive because when Eq. (2.6) is written in its equivalent format above, the former R is identical 1 1 1 1 2 2 to the later R. Both R are now just given by 1 (2n) or 1 (2n − 1) . 2 1 2 2 1 2 2 t + 4 2 t + 4 Remark 4. Whereas for Gram[y=0] points F(n) Eq. (2.6) that exactly represent precise Areas of Varying Loops and f(n) Eq. (2.5) [when interpreted as Riemann sum] that exactly represent approximate Areas of Varying Loops in a proportionate manner; so must the associated scaled amplitude R in Eq. (2.6) which is dependent on parameter t and Eq. (2.5) which is independent of parameter t represent [in a surrogate manner] corresponding precise and approximate Areas of Varying Loops in a proportionate manner. Remark 5. Whereas for Gram[x=0] points F(n) Eq. (2.8) that exactly represent precise Areas of Varying Loops and f(n) Eq. (2.7) [when interpreted as Riemann sum] that exactly represent approximate Areas of Varying Loops in a proportionate manner; so must the associated scaled amplitude R in Eq. (2.8) which is dependent on parameter t and Eq. (2.7) which is independent of parameter t represent [in a surrogate manner] corresponding precise and approximate Areas of Varying Loops in a proportionate manner. 1 Finally, we analyze f(n) = 0 and F(n) = 0 equations at σ = 2 critical line for NTZ situation where phase 1 1 shift α = π in NTZ f(n) Eq. (2.1) and − π in NTZ F(n) Eq. (2.3); and F(n) = 0 equations at σ = 1 critical 4 4 2 1 3 line for Gram[y=0] points and Gram[x=0] points situations where phase shift α = − π (or π when written 4 4 3 in its equivalent format above) in Gram[y=0] points F(n) Eq. (2.6) and − π in Gram[x=0] points F(n) Eq. 4 1 (2.8). Always being π out-of-phase with each other, trigonometric functions sine and cosine are cofunctions 2 π π π π d(sinn) d(cosn) with sin n = cos ( - n) or cos (n - ), cos n = sin ( - n) or sin (n + ), = cosn, = −sinn, 2 2 2 2 dn dn π π R sinn · dn = −cosn +C [= sin (n - ) + C] and R cosn · dn = sinn +C [= cos (n - ) + C]. Last two integrals 2 2 explain relation between f(n)’s Zeroes and F(n)’s Pseudo-zeroes when they involve simple sine and/or cosine 1 1 terms viz, f(n)’s CP Zeroes = F(n)’s CP Pseudo-zeroes – π with CP Zeroes and CP Pseudo-zeroes being π 2 2 out-of-phase with each other. Remark 6. Involving trigonometric functions as complex sine and/or cosine terms: f(n)’s IP NTZ or [non- existent] f(n)’s IP virtual NTZ (in t values) = F(n)’s IP Pseudo-NTZ or [non-existent] F(n)’s IP virtual Pseudo- 1 NTZ (in t values) – π; f(n)’s IP Gram[y=0] points or f(n)’s IP virtual Gram[y=0] points (in t values) = F(n)’s 2 3 IP Pseudo-Gram[y=0] points or F(n)’s IP virtual Pseudo-Gram[y=0] points (in t values) – π; and f(n)’s IP 4 Gram[x=0] points or f(n)’s IP virtual Gram[x=0] points (in t values) = F(n)’s IP Pseudo-Gram[x=0] points or 3 F(n)’s IP virtual Pseudo-Gram[x=0] points (in t values) – π. 4 R f (n)dn = F(n) +C where F0(n) = f (n). f(n) and F(n) are literally [connected] bijective (both injective and surjective or a one-to-one correspondence) functions. Underlying f(n) as equation and F(n) as law (equation) that generate their CIS of IP Zeroes, IP virtual Zeroes, IP Pseudo-zeroes and IP virtual Pseudo- 1 3 zeroes are precisely related as π (for NTZ case) or π (for Gram[y=0] points and Gram[x=0] points cases) 2 4 out-of-phase with each other. Peculiar to IP NTZ as Origin intercept points, we crucially note only they will uniquely behave in accordance with complex sine and/or cosine terms present in their equations that generate 1 corresponding IP Zeroes and IP Pseudo-zeroes which are π out-of-phase with each other. 2 1 The x-axis and y-axis are orthogonal to each other with angle between them = π radian. Involving trigono- 2 metric functions as complex sine and/or cosine terms: f(n)’s IP Gram[y=0] points or f(n)’s IP virtual Gram[y=0] 1 points (in t values) = f(n)’s IP Gram[x=0] points or f(n)’s IP virtual Gram[x=0] points (in t values) + π; and 2 10 Ting

F(n)’s IP Pseudo-Gram[y=0] points or F(n)’s IP virtual Pseudo-Gram[y=0] points (in t values) = F(n)’s IP 1 Pseudo-Gram[x=0] points or F(n)’s IP virtual Pseudo-Gram[x=0] points (in t values) + π. 2 Remark 7. These observations imply underlying f(n) as equation and F(n) as law (equation) that generate corresponding paired IP two types of Gram points [as Zeroes], paired IP two types of virtual Gram points [as virtual Zeroes], paired IP two types of Pseudo-Gram points [as Pseudo-zeroes], and paired IP two types of 1 virtual Pseudo-Gram points [as virtual Pseudo-zeroes] being π out-of-phase with each other. 2

3 The Completely Predictable and Incompletely Predictable entities

Cardinality of a given set: With increasing size, arbitrary Set X can be countable finite set (CFS), countable infinite set (CIS) or uncountable infinite set (UIS). Cardinality of Set X, |X|, measures number of elements in Set X. E.g. Set negative Gram[y=0] point has CFS of negative Gram[y=0] point with |negative Gram[y=0] point| = 1, Set even Prime number has CFS of even Prime number with |even Prime number| = 1, Set Natural numbers has CIS of Natural numbers with |Natural numbers| = ℵ0, and Set Real numbers has UIS of Real numbers with |Real numbers| = c (cardinality of the continuum). The word “number” [singular noun] or “numbers” [plural noun] used in reference to prime and composite numbers, nontrivial zeros and two types of Gram points can interchangeably be replaced with the word “entity” [singular noun] or “entities” [plural noun]. We outline an innovative method to classify certain appropriately chosen equations or algorithms in two ways by using relevant locational properties of its output. This output consist of generated entities either from function-based equations or from algorithms. Our classification system is formalized by providing definitive definitions for Completely Predictable (CP) entities obtained from CP equations or algorithms, and Incompletely Predictable (IP) entities obtained from IP equations or algorithms. CP simple equation or algorithm generates CP numbers. A generated CP number is locationally defined as a number whose position is independently determined by simple calculations without needing to know related positions of all preceding numbers in neighborhood. IP complex equation or algorithm generates IP numbers. A generated IP number is locationally defined as a number whose position is dependently determined by complex calculations with needing to know related positions of all preceding numbers in neighborhood. Container is a useful analogical term that metaphorically group CP entities (e.g. even and odd numbers) and IP entities (e.g. nontrivial zeros, prime and composite numbers) to be exclusively located in, respectively, Simple Container and Complex Container. Gram points and virtual Gram points are generally t-valued transcendental numbers. Simple properties are inferred from a sentence such as “This simple equation or algorithm by itself will intrinsically incorporate actual location [and actual positions] of all CP numbers”. Examples: simple equations E = (2 X i) and O = (2 X i) - 1 for i = all real numbers ≥ 0 or i = all integers ≥ 0 will respectively and intrinsically incorporate or generate CIS of all CP even number Ei = 0, 2, 4, 6,... and CIS of all CP odd numbers Oi = 1, 3, 5, 7,... whereby even number (n) is defined as “Any integer that can be divided exactly by 2 with last digit always being 0, 2, 4, 6 or 8” and odd number (n) is defined as “Any integer that cannot be divided exactly by 2 with last digit always being 1, 3, 5, 7 or 9”. Congruence n = 0 (mod 2) holds for even n and congruence n = 1 (mod 2) holds for odd n. We note the zeroth even number is given by E0 = 0. Complex properties, or meta-properties, are inferred from a sentence such as “This complex equation or algorithm by itself will intrinsically incorporate actual location [but not actual positions] of all IP numbers”. Examples: complex algorithms Pi+1 = Pi + pGapi and Ci+1 = Ci + cGapi for i = 1, 2, 3,..., ∞ with P1 = 2 and C1 = 4 will respectively and intrinsically incorporate CIS of all IP prime number 2, 3, 5, 7,... and CIS of all IP composite numbers 4, 6, 8, 9,... whereby prime numbers are defined as “All Natural numbers apart from 1 that are evenly divisible by itself and by 1” and composite numbers are defined as “All Natural numbers apart from 1 that are evenly divisible by numbers other than itself and 1”. E.g. via computed Pseudo-zeroes that can 1 be converted to Zeroes at σ = 2 critical line, complex equation DSPL will intrinsically incorporate the CIS of all IP nontrivial zeros [given as t values rounded off to six decimal places]: 14.134725, 21.022040, 25.010858, 30.424876, 32.935062, 37.586178,... and complex equation Gram[y=0] points-DSPL will intrinsically incorporate the CIS of all IP Gram[y=0] points [given as t values rounded off to six decimal places]: 0, 3.436218, Riemann hypothesis and closely related two types of Gram points 11

9.666908, 17.845599; 23.170282, 27.670182,.... Gram[y=0] points or ‘usual’ Gram points are named after Danish mathematician Jørgen Pedersen Gram (June 27, 1850 - April 29, 1916) – see [1], p. 44-45.

4 The exact and inexact Dimensional analysis homogeneity for Equations

For ‘base quantities’ length, mass and time; their fundamental SI ‘units of measurement’ meter (m) is defined as distance travelled by light in vacuum for time interval 1/299 792 458 s with speed of light c = 299,792,458 ms−1, kilogram (kg) is defined by taking fixed numerical value Planck constant h to be 6.626 070 15 X 10−34 2 −1 133 Joules·second (Js) [whereby Js is equal to kgm s ] and second (s) is defined in terms of ∆vCs = ∆( Cs)h f s = 9,192,631,770 s−1. Derived SI units such as J and ms−1 respectively represent ‘base quantities’ energy and velocity. ‘Dimension’ is commonly used to indicate ‘units of measurement’ in well-defined equations. DA is a traditional analytic tool with DA homogeneity and DA non-homogeneity (respectively) denoting valid and invalid equation occurring when ‘units of measurements’ for ‘base quantities’ are “balanced” and “unbalanced” across both sides of equation. E.g. equation 2 m + 3 m = 5 m is valid but equation 2 m + 3 kg = 5 ‘m·kg’ is invalid (respectively) manifesting DA homogeneity and non-homogeneity. We conveniently adopt concepts from DA with this justifiable action being mathematically correct and valid. Let (2n) and (2n-1) be ‘base quantities’ in DSPL formatted in simplest forms as equations. E.g. DA on 1 1 2 1 exponent 2 in (2n) when depicted in simplest form is desirable for our purpose but DA on exponent 4 in 1 equivalent (22n2) 4 not depicted in simplest form is undesirable for our purpose. Fractional exponents as ‘units 1 of measurement’ given by (1−σ) for equations when σ = 2 coincide with exact DA homogeneity; and (1−σ) 1 for equations when σ 6= 2 coincide with inexact DA homogeneity. Respectively, exact DA homogeneity at 1 σ = 2 denotes ∑(all fractional exponents) as 2(1−σ) equates to [exact] integer 1; and inexact DA homogeneity 1 at σ 6= 2 denotes ∑(all fractional exponents) as 2(1 − σ) equates to [inexact] fractional number 6=1 [Range: 0 < 2(1 − σ) < 1 and 1 < 2(1 − σ) < 2]. Computations based on exact and inexact DA homogeneity in DSPL 1 explicitly give rise to σ = 2 critical line Gram points (given as Pseudo-zeroes t-values which can be converted 1 to Zeroes t-values) and σ 6= 2 non-critical lines virtual Gram points (given as virtual Pseudo-zeroes t-values which can be converted to virtual Zeroes t-values). For calculations involving 2(1 − σ) above or 2(−σ) below, it is inconsequential whether σ values in these fractional exponents are depicted in simplest form or not in simplest form. Performing exact and inexact DA homogeneity on sim-η(s) is valid. With same ‘base quantities’, fractional exponents as ‘units of measurement’ 1 are now given by (−σ). Respectively, exact DA homogeneity at σ = 2 denotes ∑(all fractional exponents) as 1 2(−σ) equates to [exact] integer −1; and inexact DA homogeneity at σ 6= 2 denotes ∑(all fractional exponents) as 2(−σ) equates to [inexact] fractional number 6=–1 [Range: –2 < 2(−σ) < –1 and –1 < 2(−σ) < 0]. 1 Computations using sim-η(s) [when interpreted as Riemann sum] explicitly give rise to σ = 2 critical line 1 Gram points (given as Zeroes t-values) while representing exact DA homogeneity and σ 6= 2 non-critical lines virtual Gram points (given as virtual Zeroes t-values) while representing inexact DA homogeneity.

5 Gauss Areas of Varying Loops and Principle of Maximum Density for Integer Number Solutions

Legend: C = UIS complex numbers, R = UIS real numbers, Q = CIS rational numbers that include fractional numbers and rational roots, R-Q = UIS total irrational numbers, A = CIS algebraic numbers, R-A = UIS transcendental irrational numbers, Z = CIS integers which are literally fractional numbers with denominator 1, W = CIS whole numbers, N = CIS natural numbers, E = CIS even numbers, O = CIS odd numbers, P = CIS prime numbers, and C = CIS composite numbers. CIS N = Set E [whereby we did not include the zeroth even number E0 = 0] + Set O; CIS N = CIS P + CIS C + CFS Number 1; and CIS N ⊂ CIS W ⊂ CIS Z ⊂ CIS Q ⊂ UIS R ⊂ UIS C. CIS A as C (including R) = CIS Q that include fractional numbers and rational roots + CIS irrational roots whereby both rational and irrational roots are derived from non-zero polynomials. The following refined definitions are useful: UIS total irrational numbers = CIS irrational roots (numbers) + UIS transcendental irrational numbers whereby transcendental irrational numbers [algebraic] irrational 12 Ting numbers. Whereas CIS rational roots (numbers), CIS irrational roots (numbers) and UIS transcendental numbers are treated separately as mutually exclusive numbers; so must the existing algebraic functions that generate CIS rational roots (numbers) and CIS irrational roots (numbers), and the existing transcendental functions that generate UIS transcendental numbers be treated separately as mutually exclusive functions. An algebraic function [such as rational functions, square root, cube root function, etc] satisfies a polynomial equation. A transcendental function [such as exponential function, natural logarithm, trigonometric functions, hyperbolic functions, gamma, elliptic, zeta functions, etc] is an analytic function that does not sat- isfy a polynomial equation. Thus, a transcendental function “transcends” algebra since it cannot be expressed in terms of a finite sequence of algebraic operations consisting of addition, subtraction, multiplication, divi- sion, powers, and fractional powers or root extraction. All integers, rational numbers, rational or irrational 2 roots of real√ and complex numbers are algebraic numbers e.g. a root of polynomial x − x − 1 = golden ratio 1 + 5 √2 √ 1 √3 1 ϕ = = 1.618033..., square root of 2 viz, 2 or 2 = 2 2 = 1.414213..., or cube root of 2 viz, 2 = 2 3 2 ≈1.259921. Real and complex numbers that are not algebraic numbers e.g. π and e are transcendental numbers. However, we note sine and cosine as transcendental functions generally give rise to mutually exclusive√ sets of π 2π 1 1 transcendental numbers except at discrete points such as sin = sin30◦ = cos = cos60◦ = = [viz, 6 6 2 2 transcendental functions generating an algebraic number as rational root (number) at certain discrete points]. √ 1 √ Following treatise involve infinite series. An interesting property of irrational number 2 is √ = 2 + 1 2 − 1 √ √ √ since 2 + 1 2 − 1 = 2 − 1 = 1. This is related to the property of silver ratios. 2 can also be expressed in terms of copies of imaginary unit i using only square root√ and√ arithmetic√ operations,√ if the square root i + i i −i − i −i √ √ √ √ symbol is interpreted suitably for complex numbers i and -i: and i+i i and −i−i −i . i −i i −i 1 √ 1 q 2 − 2 1 Multiplicative inverse (reciprocal) of (2) or 2 is (2) or 2 which is a unique [irrational number] con- q √ π π π stant since 1 = √1 = 1 2 = cos = sin . Transcendental numbers such as (given by Leibniz series 2 2 2 4 4 4 1 1 1 1 1 π2 1 1 1 − + − + − ···≈ 0.78539816); and (given by ζ(2) = + + + ··· ≈ 1.6449340668482), 1 3 5 7 9 6 12 22 32 respectively, encode complete set of alternating odd and, by default, alternating even numbers; and natu- ∞ (−1)n+1 ral numbers. Also known as alternating zeta function, Dirichlet eta function η(s) = ∑ s when ex- n=1 n panded, will intrinsically encode complete set of alternating natural numbers e.g. η(1) = ln(2) (given by ∞ −1n+1 ∞ 1 1 1 1 1 1 1 ∑ = ∑ n [ζ(n) − 1] + = − + − + − ···≈ 0.69314718056. Equivalent Euler prod- n=1 n n=2 2 2 1 2 3 4 5 uct formula for ζ(s) with product over prime numbers [instead of summation over natural numbers] will intrinsically encode complete set of prime and, by default, composite numbers. As a side note, complete set of alternating prime and, by default, alternating composite numbers is encoded in converging alternating series ∞ k (−1) t h ∑ ≈ −0.2696063519 (transcendental number) when fully expanded whereby pk is k prime number. k=1 pk Equation of a circle centered at Origin with radius r and precise Area = πr2 is given in Cartesian coordinates as x2 + y2 = r2. The number of integer lattice points N(r) on and inside a circle [viz, pairs of integers (m,n) such that m2 + n2 ≤ r2] can be exactly determined by following two equations whereby N(r) is considered the most accurate surrogate marker of approximate Area for a given circle. Named after German mathematician Carl Friedrich Gauss (April 30, 1777 - February 23, 1855), Gauss Circle Problem is the problem of determining how many integer lattice points as approximate Area for a given circle. For i and r = 0, 1, 2, 3,...,∞ in terms of a ∞ r2 r2 sum involving the floor function, N(r) can be expressed as equation N(r) = 1 + 4 ∑ − i=0 4i + 1 4i + 3 whereby it is a consequence of Jacobi’s two-square theorem which follows almost immediately from the Jacobi triple product. A much simpler sum appears if sum of squares function r2(n) that is defined as number of ways Riemann hypothesis and closely related two types of Gram points 13

r2 of writing number n as sum of two squares is used. Then, we have alternative equation N(r) = ∑ r2(n). The n=0 first few N(r) values for r = 0, 1, 2, 3, 4, 5, 6, 7, 8,... are 1, 5, 13, 29, 49, 81, 113, 149,... whereby these are IP entities complying with relationship: [simple] equation for precise Area of circle = πr2 is proportional to above two most accurate and equivalent [complex] equations for approximate Area of circle = N(r). 2 ∞ r2(x ) 2 r2(n) √ The identity N(x) − = πx + x ∑ √ J1(2πx n) is implicitly related to number of integer lattice 2 n=1 n points, N(r), where J1 denotes Bessel function of first kind with order 1. It was first discovered by English mathematician Godfrey Harold Hardy (February 7, 1877 - December 1, 1947)[2]. 1 N(r) 1 1 1 1 1 1 1 N(r) is closely connected with Leibniz series since − = − + − + − ··· ± 4 r2 r2 1 2 3 4 5 7 1 E(r) 1 1 E(r) 1 1 1 E(r) ± = [π +2Φ(−1,1, +r)]± = [π +ψ ( (3+2r))−ψ ( (1+2r))]± , where E(r) is r r 4 2 r 4 0 4 0 4 r an error term, Φ(z,s,a) is a Lerch transcendent and ψ0(x) is a digamma function, so taking the limit r −→ ∞ π 1 1 1 1 1 √ gives = − + − + − ···. Gauss showed N(r) = πr2 + E(r), where |E(r)| ≤ 2 2πr. 4 1 2 3 4 5 Primitive Circle Problem as least accurate surrogate marker of approximate Area for a given circle involves calculating the number of coprime integer solutions (m,n) to the inequality m2 + n2 ≤ r2. If the number of such solutions is denoted V(r) then the values of V(r) for r taking small integer values are 0, 4, 8, 16, 32, 48, 72, 88, 120, 152, 192,.... Using the same ideas as usual Gauss Circle Problem and the fact that probability two integers 6 6 are coprime is , it is relatively straightforward to show V(r) = r2 + O(r1+ε ). We solve problematic part π2 π of Primitive Circle Problem by reducing the exponent in the error term. This exponent is presently best known to be 221/304 + ε when we validly assume Riemann hypothesis to be true in this paper. Remark 8. Let A denote Area of a given circle with radius r. The computed precise A using A = πr2 method, computed approximate A using [most accurate] approximate N(r) method of Gauss Circle Problem and computed approximate A using [least accurate] approximate A(r) method of Primitive Circle Problem will explicitly confirm A ∝ r2 for all three methods. 1 1 Lemma 1. We can validly and fully demonstrate that only when σ = 2 [and not when σ 6= 2 ] in sim-η(s) or DSPL will the maximum number of integer solutions (constituted by all integers ≥ 1) arise that must uniquely comply with Principle of Maximum Density for Integer Number Solutions. Proof. We translate concepts from Gauss Circle Problem and Primitive Circle Problem onto Gauss Areas of Varying Loops. This exercise fully support all below materials in this section. For n classically involving all integers ≥1 in sim-η(s) as ∆n −→ 1 or n classically involving all real numbers ≥1 in DSPL as ∆n −→ 0; their base quantities (2n) and (2n-1) will, respectively, generate CIS even numbers commencing from 2 and CIS odd numbers commencing from 1. These base quantities are subjected to algebraic function square roots 1 1 1 2 at σ = 2 critical line [viz, when σ = 2 ] and cube roots at σ = 3 non-critical line or twice cube roots at σ = 3 1 non-critical line [viz, when σ 6= 2 ] thus giving rise to corresponding subset of rational roots and subset of irrational roots. Relevant to Remark 9, we now concentrate on combined (2n)’s and (2n-1)’s obtained integer lattice points [≥ 1] to derive solitary subset of rational roots for n = 1 to 100 range in sim-η(s) or DSPL when: 1 (I) σ = 2 involving a neither even nor odd function with no symmetry viz, f (−n) 6= f (n) and f (−n) 6= − f (n) 1 by applying f (n) as fractional exponent 2 or square root on n = ten perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 giving rise to (maximum) ten rational roots as consecutive integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1 (II) σ = 3 involving a odd function with Origin symmetry viz, f (−n) = − f (n) by applying f (n) as fractional 1 exponent 3 or cube root on n = four perfect cubes 1, 8, 27, 64 giving rise to (non-maximum) four rational roots as consecutive integer solutions 1, 2, 3, 4. 2 (III) σ = 3 involving an even function with y-axis symmetry viz, f (−n) = f (n) by applying f (n) as fractional 2 exponent 3 or squared cube root on n = four perfect cubes 1, 8, 27, 64 giving rise to (non-maximum) four rational roots as non-consecutive integer solutions 1, 4, 9, 16. 1 1 Remark 9. Only at σ = 2 critical line which involves applying f (n) as fractional exponent 2 or square root on n = all perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... will we obtain maximum number of rational roots 14 Ting as consecutive integer solutions 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... (viz, all integers ≥1). This observation uniquely 1 comply with Principle of Maximum Density for Integer Number Solutions at σ = 2 critical line. The proof is now complete for Lemma 12.

The sim-η(s) or DSPL is classified as complex function or law with single variable n and parameters σ, t. All their derived equations [Eqs. (2.1) to (2.8)] have (2n)- or (2n-1)-complex term with algebraic functions consisting of powers, fractional powers, root extraction (equating to scaled amplitude R in Remarks 3, 4 and 5) [that are dependent on parameter σ] and (2n)- or (2n-1)-complex term with transcendental functions consisting of sine, cosine, single cosine wave, single sine wave, natural logarithm [that are independent of parameter σ]. Abbreviations: Term-(2n) = (2n)-complex term with algebraic functions X (2n)-complex term with transcendental functions; and Term-(2n-1) = (2n-1)-complex term with algebraic functions X (2n-1)-complex term with transcendental functions. Remark 10. Corresponding to Areas of Varying Loops = 0 in f(n) sim-η(s) or F(n) DSPL, Term-(2n) must 1 precisely cancel Term-(2n-1) in order to obtain σ = 2 f(n)’s Zeroes and F(n)’s Pseudo-zeroes or to obtain 1 σ 6= 2 f(n)’s virtual Zeroes and F(n)’s virtual Pseudo-zeroes. In sim-η(s) or DSPL, its computed CIS rational roots (subset) as integers [rational numbers] + computed CIS irrational roots (subset) as irrational numbers = computed CIS total roots. Remark 11. Complex function F(n) = DSPL [representive of precise Area under the Curve] will generate the most accurate precise Areas of Varying Loops [when all rational and irrational roots from combined base quantities (2n) and (2n-1) are utilized] and the least accurate precise Areas of Varying Loops [when only rational roots from combined base quantities (2n) and (2n-1) are utilized]; and complex function f(n) = sim- η(s) when interpreted as Riemann sum [representive of approximate Area under the Curve] will generate the most accurate approximate Areas of Varying Loops [when all rational and irrational roots from combined base quantities (2n) and (2n-1) are utilized] and the least accurate approximate Areas of Varying Loops [when only rational roots from combined base quantities (2n) and (2n-1) are utilized]. Our [metaphoric] varying radius r in sim-η(s) or DSPL is defined as r = Term-(2n) – Term-(2n-1) whereby 1 perpetually recurring r = 0 will correspond to Areas of Varying Loops = 0 in order to obtain σ = 2 f(n)’s 1 Zeroes and F(n)’s Pseudo-zeroes or to obtain σ 6= 2 f(n)’s virtual Zeroes and F(n)’s virtual Pseudo-zeroes. In effect, Areas of Varying Loops is conceptionally synonymous with varying radius r whereby varying radius r can also be perceived as [metaphoric] varying distance between Term-(2n) and Term-(2n-1). Remark 12. Whether involving the most accurate method using total roots or the least accurate method using rational roots to determine DSPL’s precise or sim-η(s)’s approximate Areas of Varying Loops, we can explicitly conclude all these (infinitely many) obtained Areas of Varying Loops ∝ [and =] varying radius r will repeat in a perpectually dynamic, cyclical and IP manner.

6 Simple observation on Shift of Varying Loops in ζ(σ + ıt) Polar Graph and Principle of Equidistant for Multiplicative Inverse

For single-term trigonometric function f(n) = sin(n), it is an odd function with Origin symmetry since -f(n) = f(-n) for all n. The f(n) = sin(n) has an infinite number of CP x-axis intercept points (Zeroes) and a solitary unique Origin intercept point (Zero) since it belong to a class of odd functions that is defined at n = 0 and must 1 pass through the Origin. Otherwise, the other class of odd functions such as f(n) = sin( n ) with infinite number 1 of CP x-axis intercept points (Zeroes) but without Origin intercept point [since sin( n is undefined at n = 0] can remain symmetrical about the Origin without actually passing through it. For single term trigonometric function f(n) = cos(n) with symmetry about the y-axis, it is an even function since f(n) = f(-n) for all n. It has an infinite number of CP x-axis intercept points (Zeroes). Being undefined at n = 0, it will never have Origin intercept point. For dual terms trigonometric functions f(n) = cos(n) - sin(n) and f(n) = cos(n) + sin(n), they are neither even nor odd without any symmetry. They both have an infinite number of CP x-axis intercept points (Zeroes). Being undefined at n = 0, they will never have Origin intercept point. Special properties are given below for Addition Riemann hypothesis and closely related two types of Gram points 15 and Multiplication: The sum or difference of two even functions is even. The sum or difference of two odd functions is odd. The sum or difference of an even and odd function is neither even nor odd unless one function is zero; viz, there is (exactly) one function that is both even and odd, and it is the zero function f(n) = 0. The product of two even functions is an even function. The product of two odd functions is an even function. The product of an even function and an odd function is an odd function. Trigonometric identity for linear combination of sine and cosine function: Here, we use the notation f(x) instead of f(n). The trigonometric identity for linear combination of sine and cosine acos(x) + bsin(x) can be freely, arbitrarily and interchangeably written as either [simple] single cosine wave Rcos(x−α)√or [simple] single sine wave Rsin(x + α) whereby R is the scaled amplitude and α is the phase shift. R = a2 + b2 = 1 b a b (a2 + b2) 2 . Since sin(α) = √ and cos(α) = √ , then α = tan−1 . Below, we conveniently a2 + b2 a2 + b2 a √ 1 assign 2 to equivalently denote 2 2 . √ √ 1 √ 3 With a = 1, b = -1, R = 2; cos(x) − sin(x) = 2cos x + π = 2sin x + π . 4 4 √ √ 1 √ 3 With a = -1, b = 1, R = 2; −cos(x) + sin(x) = 2sin x − π = 2cos x − π . 4 4 √ √ 1 √ 1 With a = 1, b = 1, R = 2; cos(x) + sin(x) = 2cos x − π = 2sin x + π . 4 4 √ √ 3 √ 3 With a = -1, b = -1, R = 2; −cos(x) − sin(x) = 2cos x + π = 2sin x − π . 4 4 R f (x)dx = F(x) +C with F0(x) = f (x). With |a| = 1 and |b| = 1, consider single-term [simple] trigonometric functions: f (x) = acos(x) which belongs to an even function and f (x) = bsin(x) which belongs to an odd function. Whereas all linear combination of [simple] cos(x) and [simple] sin(x) as sum or difference such as f (x) = cos(x) + sin(x) and f (x) = cos(x) - sin(x) belong to neither even nor odd functions, then their corresponding F(x) being linear combination of [simple] cos(x) and [simple] sin(x) as sum or difference must also belong to neither even nor odd functions. With both f(x) and corresponding F(x) considered as [simple] functions and relevant trigonometric identities being applied, they can intrinsically and arbitrarily be expressed 1 3 as either [simple] single cosine wave or [simple] single sine wave containing a phase shift π or π and a 4 4 √ 1 2 1 scaled amplitude 2 [= 2 which is base 2 endowed with exponent 2 ]. Respectively, F(x) and f(x) have an infinite number of x-axis intercept points called Pseudo-zeroes and Zeroes but with nil Origin intercept point. We reiterate that Gram[x=0,y=0] points, Gram[y=0] points and Gram[x=0] points are three types of IP 1 Gram points [Zeroes] occurring at σ = 2 critical line (Figure 2) based on, respectively, Origin intercept points, x-axis intercept points and y-axis intercept points. They can be dependently computed from η(s) [proxy for ζ(s)] and sim-η(s) whereby sim-η(s) is obtained by applying Euler formula to η(s). Gram[x=0,y=0] points are synonymous with NTZ and Gram[y=0] points are synonymous with ‘usual’ Gram points. Virtual Gram[y=0] 1 points and virtual Gram[x=0] points are two types of IP virtual Gram points [virtual Zeroes] occurring at σ 6= 2 non-critical lines based on, respectively, x-axis intercept points and y-axis intercept points – see Figure 3 for 2 3 σ = 5 and Figure 4 for σ = 5 . They can be dependently computed from these exact same functions. Lemma 2. Both sim-η(s) and DSPL manifest Principle of Equidistant for Multiplicative Inverse. 1 Proof. Let δ = 10 . This will generate in Figure 3 and Figure 4 the δ induced shift of [infinite] Varying Loops 1 in reference to Origin; viz, the simple relationship of [more negative] left-shift given by ζ( 2 − δ + ıt) [Figure 1 1 3] < [neutral] nil-shift given by ζ( 2 + ıt) [Figure 2] < [more positive] right-shifted given by ζ( 2 + δ + ıt) [Figure 4] will always be consistently true. 1 1 1 Given δ = 10 , the σ = 2 − δ non-critical line (represented by Figure 3) and σ = 2 + δ non-critical line 1 (represented by Figure 4) are equidistant from σ = 2 critical line (represented by Figure 2). We refer to the important comment that precede Remark 10. Then additive inverse operation of sin(δ) + sin(-δ) = 0 indicating symmetry with respect to Origin [or cos(δ) - cos(-δ) = 0 indicating symmetry with respect to y-axis] is not applicable to our complex single sine wave [or single cosine wave] since (2n)- or (2n-1)-complex term with transcendental functions consisting of sine, cosine, single sine wave, single cosine wave, natural logarithm 16 Ting are independent of parameter σ. However, (2n)- or (2n-1)-complex term with algebraic functions consisting of powers, fractional powers, root extraction (equating to scaled amplitude R in Remarks 3, 4 and 5) 1 1 are dependent on parameter σ. Let x = (2n) or or (2n − 1) or . With multiplicative inverse (2n) (2n − 1) 1 1 operation of xδ ·x−δ = 1 or · = 1 that is applicable, this imply intrinsic presence of Multiplicative Inverse xδ x−δ in sim-η(s) or DSPL for all σ values with this function or law rigidly obeying relevant trigonometric identity. Remark 13. We named the above-mentioned phenomenon Principle of Equidistant for Multiplicative 1 Inverse. We additionally note by letting δ = 0, we always generate Figure 2 that represents σ = 2 critical line. The proof is now complete for Lemma 22.

Zeroes and Pseudo-zeroes: There are three types of stationary points in a given periodic f(n) involving sine and/or cosine functions that could act as x-axis intercept points via three types of f(n)’s Zeroes with corresponding three types of F(n)’s Pseudo-zeroes: maximum points e.g. with f(n) or F(n) = sin n - 1; minimum points e.g. with f(n) or F(n) = sin n + 1; and points of inflection e.g. with f(n) or F(n) = sin n [which also has Origin intercept point as a Zero or Pseudo-zero]. A fourth type of f(n)’s Zeroes and F(n)’s Pseudo-zeroes consist of non-stationary points occurring e.g. with f(n) or F(n) = sin n + 0.5. With (j - i) = (l - k) = 2π [viz, one Full cycle], let a given Zero be located in f(n)’s interval [i,j] viz, i < Zero < j; and its corresponding Pseudo-zero be located in F(n)’s Pseudo-interval [k,l] viz, k < Pseudo-zero < l. For this Zero and Pseudo-zero characterized by either point of inflection or non-stationary point; both will comply with preserving positivity [going from (-ve) below x-axis to (+ve) above x-axis] as explained using the Zero case [with the Pseudo-zero case following similar lines of explanations]. This can be stated as follow for interval [i,j]: If j > i, then computed f(j) > computed f(i). In particular, the condition “If i ≥ 0, then computed f(i) ≥ 0” must not be present for these two particular types of Zero to validly exist in interval [i,j]. With reversal of inequality signs, the converse situation for j < Zero < i and corresponding l < Pseudo-zero < k will be equally true in preserving negativity [going from (+ve) above x-axis to (-ve) below x-axis]. All these are useful observed properties for Zeroes and Pseudo-zeroes. Preservation or conservation of Net Area Value and Total Area Value with definitions[1], p. 10 - 13: R f (n)dn = F(n) +C with F0(n) = f (n). Consider a nominated function f (n) for interval [a,b]. We define Net Area Value (NAV) calculated using its antiderivative F(n) as the net difference between positive area value(s) [above horizontal x-axis] and negative area value(s) [below horizontal x-axis] in interval [a,b]; viz, NAV = all +ve value(s) + all -ve value(s). Again calculated using F(n), we define Total Area Value (TAV) as the total sum of (absolute value) positive area value(s) [above horizontal x-axis] and (absolute value) negative area value(s) [below horizontal x-axis] in interval [a,b]; viz, TAV = all |+ve value(s)| + all |-ve value(s)|. Calculated NAV and TAV are precise using antiderivative F(n) obtained from integration of f (n) but are only approximate when using Riemann sum on f (n). For f(n)’s interval [a,b] whereby a = initial Zero and b = next Zero, and F(n)’s Pseudo interval [c,d] whereby c = initial Pseudo-zero and d = next Pseudo-zero; then compliance with preservation or conservation of NAV and TAV will simultaneously occur in both f(n)’s Zeroes and F(n)’s Pseudo-zeroes given by their sine and/or cosine functions only when Zero gap = (b - a) = Pseudo-zero gap = (d - c) = 2π [viz, involving one Full cycle]. For our purpose, NAV = 0 condition is validly preserved or conserved 1 for f(n) sim-η(s)’s IP Zeroes and F(n) DSPL’s IP Pseudo-zeroes at parameter σ = 2 . Ditto for f(n) sim-η(s)’s 1 IP virtual Zeroes and F(n) DSPL’s IP virtual Pseudo-zeroes at parameter σ 6= 2 ; viz, NAV = 0 condition is validly preserved or conserved for f(n) sim-η(s)’s IP virtual Zeroes and F(n) DSPL’s IP virtual Pseudo-zeroes. For all our complex functions and complex equations in this paper, s = σ ±it whereby one would commonly or routinely invoke s = σ + it for discussion. For all f(n) and F(n) general equations depicted below without trigonometric identity application, we note the presence of mixed sine and cosine terms in all these general equations except for f(n)’s Gram[y=0] points-sim-η(s) and f(n)’s Gram[x=0] points-sim-η(s). I. NTZ or Gram [x=0,y=0] points as geometrical Origin intercept points are mathematically defined by ∑ReIm{η(s)} = Re{η(s)} + Im{η(s)} = 0. General equation for f(n)’s sim-η(s) as Zeroes is given by Riemann hypothesis and closely related two types of Gram points 17

∞ ∑ −(2n)−σ (sin(t ln(2n)) − cos(t ln(2n))) n=1 ∞ − ∑ −(2n − 1)−σ (sin(t ln(2n − 1)) − cos(t ln(2n − 1))) = 0 (6.1) n=1 General equation for F(n)’s DSPL as Pseudo-zeroes to Zeroes conversion is given by 1 · (2n)1−σ ((t + σ − 1)sin(t ln(2n)) + (t − σ + 1) 2(t2 + (σ − 1)2) (6.2) · cos(t ln(2n))) − (2n − 1)1−σ ((t + σ − 1) ∞ · sin(t ln(2n − 1)) + (t − σ + 1)cos(t ln(2n − 1))) +C 1 = 0 II. Gram[y=0] points as geometrical x-axis intercept points are mathematically deﬁned by ∑ReIm{η(s)} = Re{η(s)} + 0, or simply Im{η(s)} = 0. General equation for f(n)’s Gram[y=0] points-sim- η(s) as Zeroes is given by

∞ ∞ ∑ (2n)−σ sin(t ln(2n)) − ∑ (2n − 1)−σ sin(t ln(2n − 1)) = 0 (6.3) n=1 n=1 General equation for F(n)’s Gram[y=0] points-DSPL as Pseudo-zeroes to Zeroes conversion is given by − 1 · (2n)1−σ ((σ − 1)sin(t ln(2n)) +t cos(t ln(2n)))− 2(t2+(σ−1)2)

1−σ ∞ (2n − 1) ((σ − 1)sin(t ln(2n − 1)) +t cos(t ln(2n − 1))) +C 1 = 0 (6.4) III. Gram[x=0] points as geometrical y-axis intercept points are mathematically deﬁned by ∑ReIm{η(s)} = 0 + Im{η(s)}, or simply Re{η(s)} = 0. General equation for f(n)’s Gram[x=0] points-sim- η(s) as Zeroes is given by

∞ ∞ ∑ (2n)−σ cos(t ln(2n)) − ∑ (2n − 1)−σ cos(t ln(2n − 1)) = 0 (6.5) n=1 n=1 General equation for F(n)’s Gram[x=0] points-DSPL as Pseudo-zeroes to Zeroes conversion is given by 1 · (2n)1−σ (t sin(t ln(2n)) − (σ − 1)cos(t ln(2n)))− 2(t2+(σ−1)2) 1−σ ∞ (2n − 1) (t sin(t ln(2n − 1)) − (σ − 1)cos(t ln(2n − 1))) +C 1 = 0 (6.6) p y Cartesian Coordinates (x,y) is related to Polar Coordinates (r,θ) with r = x2 + y2 and θ = tan−1( ). In x anti-clockwise direction, it has four quadrants defined by the + or - of (x,y); viz, Quadrant I as (+,+), Quadrant II as (-,+), Quadrant III as (-,-), and Quadrant IV as (+,-). NTZ are Origin intercept points or Gram [x=0,y=0] points. With ‘gap’ being synonymous with ‘interval’, NTZ gap is given by initial NTZ t-value minus next NTZ t-value. Running a Full cycle from 0π to 2π, size of each IP Varying Loop in Figure 2 is proportional to magnitude of its corresponding IP NTZ varying gap. We note the 2π here as observed in Figure 2 [on Gram points], Figure 3 [on virtual Gram points] and Figure 4 [on virtual Gram points] refers to IP Varying Loops transversed by parameter t with NTZ (Gram [x=0,y=0] points) corresponding to t values as Origin intercept on Origin’s solitary (0,0) part (point); Gram [y=0] points and virtual Gram [y=0] points corresponding to t values as x-axis intercept on x-axis’ (+ve) 0π part and (-ve) 1π part; and Gram [x=0] points and virtual Gram [x=0] π 3π points corresponding to t values as y-axis intercept on y-axis’ (+ve) 2 part and (-ve) 2 part. The virtual NTZ entities do not exist; viz, Origin intercept points do not occur in Figure 3 and Figure 4. With η(s) being the proxy function for ζ(s), NTZ are mathematically defined by η(s) = 0 or sim-η(s) = 1 0. Riemann hypothesis is the original 1859-dated conjecture that all NTZ are located on σ = 2 critical line of 1 ζ(s). Mathematically proving all NTZ location on critical line as denoted by solitary σ = 2 value equates to 1 geometrically proving all Origin intercept points occurrence at solitary σ = 2 value. Both result in rigorous 18 Ting proof for Riemann hypothesis. Locations of first 10,000,000,000,000 NTZ on critical line have previously 1 been computed to be correct. Hardy[2], and with Littlewood[3], showed infinite NTZ on σ = 2 critical line by considering moments of certain functions related to ζ(s). Remark 14. The above-mentioned discovery cannot constitute rigorous proof for Riemann hypothesis 1 because they have not exclude theoretical existence of NTZ located away from this line (when σ 6= 2 ). Fur- thermore, it is literally a mathematical impossibility (“mathematical impasse”) to be able to computationally check [in a successful manner] locations of all the infinitely many NTZ to lie on critical line. The monumental task of solving Riemann hypothesis is completed by deriving F(n) DSPL from f (n) sim- η(s) with its computed Pseudo-zeroes and virtual Pseudo-zeroes which can all be converted to corresponding Zeroes and virtual Zeroes since F(n)’s IP Pseudo-zeroes and IP virtual Pseudo-zeroes (t values) = f(n)’s IP π Zeroes and IP virtual Zeroes (t values) + [for NTZ situation] whereby both f(n) and F(n) have parameters σ 2 and t. Correctly deducing exact DA homogeneity in DSPL symbolizes rigorous proof for Riemann hypothesis which is depicted as Pseudo-zeroes to Zeroes conversion that obeys relevant trigonometric identities. Three types of [traditionally] finite-interval Riemann Sums: Left / Right / Midpoint Riemann Sum uses left endpoints / right endpoints / midpoints of the subintervals. With n = 1, 2, 3,..., ∞ and therefore ∆n = 1, we note f(n) can analogically be interpreted as approximate Area under the Curve (AUC) [right infinite- ∞ ∞ 2 4 6 ∞ interval] Riemann sum ∑ f (n)∆n = ∑ f (n) = ∑ f (n) + ∑ f (n) + ∑ f (n) +...+ ∑ f (n). Corresponding n=1 n=1 n=1 n=3 n=5 n=∞−1 Z n=∞ R n=2 R n=3 solution to exact AUC improper integral f (n)dn can be validly expanded as n=1 f (n)dn + n=2 f (n)dn n=1 R n=4 R n=∞ 2 3 4 ∞ + n=3 f (n)dn +...+ n=∞−1 f (n)dn = [F(n) +C 1 + [F(n) +C 2 + [F(n) +C 3 +...+ [F(n) +C ∞−1 which, for all sufficiently large n as n−→ ∞, will manifest divergence by oscillation (viz. for all sufficiently large n as n−→ ∞, this cummulative total will not diverge in a particular direction to a solitary well-defined limit value since the [complex] sine and/or cosine terms present in sim-η(s) and DSPL are periodic transcendental-type functions). Evaluation of definite integrals Eq. (2.3) or Eq. (6.2), Eq. (2.6) or Eq. (6.4) and Eq. (2.8) or Eq. (6.6) using limit as n → +∞ for 0 < t < +∞ enable countless computations resulting in t values for (respectively) CIS NTZ, CIS Gram[y=0] points and CIS Gram[x=0] points [all as Pseudo-zeroes to Zeroes conversion]. Larger n values used for computations will correspond to increasing accuracy of these entities. Z n=∞ Remark 15. Whereas exact AUC from F(n) given by DSPL = sim − η(s)dn and approximate AUC n=1 ∞ from f(n) given by sim-η(s) = ∑ sim-η(s) [when interpreted as Riemann sum] are proportional; the Zeroes n=1 when indirectly derived from DSPL [as Pseudo-zeroes converted to Zeroes] and the Zeroes when directly 1 derived from sim-η(s) must agree with each other at σ = 2 critical line.

7 Riemann zeta function, Dirichlet eta function, simpliﬁed Dirichlet eta function and Dirichlet Sigma-Power Law

An L-function consists of a Dirichlet series with a functional equation and an Euler product. Examples of L- functions come from modular forms, elliptic curves, number fields, and Dirichlet characters, as well as more generally from automorphic forms, algebraic varieties, and Artin representations. They form an integrated com- ponent of ‘L-functions and Modular Forms Database’ (LMFDB) with far-reaching implications. In perspective, ζ(s), being the simplest example of an L-function, is a function of complex variable s (= σ ± ıt) that analyti- ∞ 1 1 1 1 cally continues sum of infinite series ζ(s) = = + + + ···. The common convention is to write ∑ ns 1s 2s 3s √ n=1 s as σ + ıt with ı = −1, and with σ and t real. Valid for σ > 0, we write ζ(s) as Re{ζ(s)}+ıIm{ζ(s)} and note that ζ(σ + ıt) when 0 < t < +∞ is the complex conjugate of ζ(σ − ıt) when −∞ < t < 0. Also known as alternating zeta function, η(s) must act as proxy for ζ(s) in critical strip (viz. 0 < σ < 1) 1 containing critical line (viz. σ = 2 ) because ζ(s) only converges when σ > 1. This implies ζ(s) is undefined to Riemann hypothesis and closely related two types of Gram points 19 left of this region in critical strip which then requires η(s) representation instead. They are related to each other 1 ∞ (−1)n+1 1 1 1 as ζ(s) = γ · η(s) with proportionality factor γ = 1−s and η(s) = ∑ s = s − s + s − ···. (1 − 2 ) n=1 n 1 2 3

∞ 1 ζ(s) = ∑ s (7.1) n=1 n 1 1 1 = + + + ··· 1s 2s 3s 1 = Π p prime (1 − p−s) 1 1 1 1 1 1 = . . . . ··· ··· (1 − 2−s) (1 − 3−s) (1 − 5−s) (1 − 7−s) (1 − 11−s) (1 − p−s)

Eq. (7.1) is deﬁned for only 1 < σ < ∞ region where ζ(s) is absolutely convergent with no zeros located here. In Eq. (7.1), equivalent Euler product formula with product over prime numbers [instead of summation over natural numbers] also represents ζ(s) =⇒ all prime and, by default, composite numbers are (intrinsically) encoded in ζ(s).

πs ζ(s) = 2sπs−1 sin ·Γ (1 − s) · ζ(1 − s) (7.2) 2

1 With σ = 2 as symmetry line of reﬂection, Eq. (7.2) is Riemann’s functional equation valid for −∞ < σ < ∞. It can be used to ﬁnd all trivial zeros on horizontal line at ıt = 0 occurring when σ = -2, -4, -6, -8, -10,. . . , ∞ πs whereby ζ(s) = 0 because factor sin( ) vanishes. Γ is gamma function, an extension of factorial function [a 2 product function denoted by ! notation whereby n! = n(n − 1)(n − 2)...(n −(n − 1))] with its argument shifted down by 1, to real and complex numbers. That is, if n is a positive integer, Γ (n) = (n − 1)!

1 ∞ (−1)n+1 ζ(s) = 1−s ∑ s (7.3) (1 − 2 ) n=1 n 1 1 1 1 = − + − ··· (1 − 21−s) 1s 2s 3s

Eq. (7.3) is defined for all σ > 0 values except for simple pole at σ = 1. As alluded to above, ζ(s) without 1 ∞ (−1)n+1 1−s viz. ∑ s is η(s). It is a holomorphic function of s defined by analytic continuation and is (1 − 2 ) n=1 n mathematically defined at σ = 1 whereby analogous trivial zeros with presence for η(s) [but not for ζ(s)] on 2πk vertical straight line σ = 1 are found at s = 1 ± ı where k = 1, 2, 3, 4,. . . , ∞. ln(2) Euler formula can be stated as eın = cosn + ı · sinn. Euler identity (where n = π) is eıπ = cosπ + ı · sinπ = −1 + 0 [or stated as eıπ + 1 = 0]. The ns of ζ(s) is expanded to ns = n(σ+ıt) = nσ et ln(n)·ı since nt = et ln(n). Apply Euler formula to ns result in ns = nσ (cos(t ln(n)) + ı · sin(t ln(n)). This is written in trigonometric form [designated by short-hand notation ns(Euler)] whereby nσ is modulus and t ln(n) is polar angle (argument). s 1 Apply n (Euler) to Eq. (7.3) to obtain f(n) general sim-η(s) for determining σ = 2 NTZ versus (non- 1 existent) σ 6= 2 virtual NTZ[1], section 4, p. 24 - 28. This is given as Eq. (6.1) and with relevant trigonometric 1 identity application [at σ = 2 ] as Eq. (2.1). Integrate f(n) general sim-η(s) to obtain F(n) general DSPL for 1 1 determining σ = 2 Pseudo-zeroes versus (non-existent) σ 6= 2 virtual Pseudo-zeroes. Pseudo-zeroes and (non- existent) virtual Pseudo-zeroes can be converted to Zeroes (NTZ) and (non-existent) virtual Zeroes (virtual 1 NTZ). This is given as Eq. (6.2) and with relevant trigonometric identity application [at σ = 2 ] as Eq. (2.3). 20 Ting

1 1 We provide f(n) general Gram[y=0] points-sim-η(s) for determining σ = 2 Gram[y=0] points versus σ 6= 2 virtual Gram[y=0] points[1], section 5, p. 28 - 30. This is given as Eq. (6.3) but we are unable to apply relevant trigonometric identity. Integrate f(n) general Gram[y=0] points-sim-η(s) to obtain F(n) general Gram[y=0] 1 1 points-DSPL for determining σ = 2 Pseudo-zeroes versus σ 6= 2 virtual Pseudo-zeroes. Pseudo-zeroes and virtual Pseudo-zeroes can be converted to Zeroes (Gram[y=0] points) and virtual Zeroes (virtual Gram[y=0] 1 points). This is given as Eq. (6.4) and with relevant trigonometric identity application [at σ = 2 ] as Eq. (2.6). 1 1 We provide f(n) general Gram[x=0] points-sim-η(s) for determining σ = 2 Gram[x=0] points versus σ 6= 2 virtual Gram[x=0] points[1], section 5, p. 28 - 30. This is given as Eq. (6.5) but we are unable to apply relevant trigonometric identity. Integrate f(n) general Gram[x=0] points-sim-η(s) to obtain F(n) general Gram[x=0] 1 1 points-DSPL for determining σ = 2 Pseudo-zeroes versus σ 6= 2 virtual Pseudo-zeroes. Pseudo-zeroes and virtual Pseudo-zeroes can be converted to Zeroes (Gram[x=0] points) and virtual Zeroes (virtual Gram[x=0] 1 points). This is given as Eq. (6.6) and with relevant trigonometric identity application [at σ = 2 ] as Eq. (2.8).

8 Conclusions

In this paper, we intrinsically treat and analyze in a de novo fashion relevant simple and complex single- variable function f(n) or F(n) and their simple and complex single-variable equation f(n) = 0 or F(n) = 0 as unique Completely Predictable or Incompletely Predictable mathematical objects. Dirichlet Sigma-Power Law symbolizes the end-product proof on Riemann hypothesis. The critical line 1 of Riemann zeta function is denoted by σ = 2 whereby all nontrivial zeros are proposed to be located in 1859 Riemann hypothesis. Treated as Incompletely Predictable problems, we gave an elementary and rigorous proof on Riemann hypothesis while also explaining existence of three types of Gram points and two types of virtual Gram points. These are achieved by analyzing complex (meta-) properties of Dirichlet Sigma-Power Law, Gram[y=0] points-Dirichlet Sigma-Power Law and Gram[x=0] points-Dirichlet Sigma-Power Law for Pseudo-Gram points; and virtual Gram[y=0] points-Dirichlet Sigma-Power Law and virtual Gram[x=0] points- Dirichlet Sigma-Power Law for virtual Pseudo-Gram points. The exact Dimensional analysis homogeneity 1 [occurring only once at σ = 2 critical line] in these Laws is endowed with ability to convert their computed Pseudo-zeroes to Zeroes resulting in nontrivial zeros (Origin intercept points or Gram[x=0,y=0] points) as one type of Gram points plus two remaining types of Gram points. The inexact Dimensional analysis homogeneity 1 [occurring inﬁnitely often at σ 6= 2 non-critical lines] in these Laws is endowed with ability to convert their computed virtual Pseudo-zeroes to virtual Zeroes resulting in two types of virtual Gram points.

Declaration of interests This work was supported by the author’s January 20, 2020 private research grant of AUS $5,000 from Mrs. Connie Hayes and Mr. Colin Webb. He received AUS $3,250 and AUS $510 reimbursements from Q-Pharm for participating in EyeGene Shingles trial and Biosimilar Study on G-CSF commencing on March 10, 2020 and February 10, 2021. Acknowledgment The author is grateful to Editor-in-Chief Professor Marcelo Viana and involved referees for carefully reading the manuscript and making useful suggestions. Many thanks to Professor Ken Ono for previous input. Dedicated to author’s daughter Jelena born 13 weeks early on May 14, 2012 and all front-line health workers globally ﬁghting against the deadly 2020 Coronavirus Pandemic.

References

1. Ting, J.Y.C. (2020). Mathematical Modelling of COVID-19 and Solving Riemann Hypothesis, Polignac’s and Twin Prime Conjectures Using Novel Fic-Fac Ratio With Manifestations of Chaos-Fractal Phenomena. J. Math. Res., 12(6) pp. 1-49. https://doi.org/10.5539/jmr.v12n6p1 2. Landau, E. Vorlesungen uber Zahlentheorie. New York: Chelsea, (2) pp. 183-308. 3. Hardy, G. H. (1914). Sur les Zeros de la Fonction ζ(s) de Riemann. C. R. Acad. Sci. Paris, 158, pp. 1012-1014. JFM 45.0716.04 Reprinted in (Borwein et al., 2008) 4. Hardy, G. H., & Littlewood, J. E. (1921). The zeros of Riemann’s zeta-function on the critical line. Math. Z., 10 (3-4), pp. 283-317. http://dx.doi:10.1007/BF01211614